envelope

0th

Percentile

Simulation envelopes of summary function

Computes simulation envelopes of a summary function.

Keywords
spatial
Usage
envelope(Y, fun=Kest, nsim=99, nrank=1, verbose=TRUE, ...,
  simulate=NULL, start=NULL, control=list(nrep=1e5, expand=1.5))
Arguments
Y
Either a point pattern (object of class "ppp") or a fitted point process model (object of class "ppm").
fun
Function that computes the desired summary statistic for a point pattern.
nsim
Number of simulated point patterns to be generated when computing the envelopes.
nrank
Integer. Rank of the envelope value amongst the nsim simulated values. A rank of 1 means that the minimum and maximum simulated values will be used.
verbose
Logical flag indicating whether to print progress reports during the simulations.
...
Extra arguments passed to fun.
simulate
Optional. An expression. If this is present, then the simulated point patterns will be generated by evaluating this expression nsim times.
start,control
Optional. These specify the arguments start and control of rmh, giving complete control over the simulation algorithm.
Details

Simulation envelopes can be used to assess the goodness-of-fit of a point process model to point pattern data. See the References.

This function first generates nsim random point patterns in one of the following ways.

  • IfYis a point pattern (an object of class"ppp") andsimulate=NULL, then this routine generatesnsimsimulations of Complete Spatial Randomness (i.e.nsimsimulated point patterns each being a realisation of the uniform Poisson point process) with the same intensity as the patternY.
  • IfYis a fitted point process model (an object of class"ppm") andsimulate=NULL, then this routine generatesnsimsimulated realisations of that model.
  • Ifsimulateis supplied, then it must be an expression. It will be evaluatednsimtimes to yieldnsimpoint patterns.
The summary statistic fun is applied to each of these simulated patterns. Typically fun is one of the functions Kest, Gest, Fest, Jest, pcf, Kcross, Kdot, Gcross, Gdot, Jcross, Jdot, Kmulti, Gmulti, Jmulti or Kinhom. It may also be a character string containing the name of one of these functions.

The statistic fun can also be a user-supplied function; if so, then it must have arguments X and r like those in the functions listed above, and it must return an object of class "fv".

Upper and lower pointwise envelopes are computed pointwise (i.e. for each value of the distance argument $r$), by sorting the nsim simulated values, and taking the m-th lowest and m-th highest values, where m = nrank. For example if nrank=1, the upper and lower envelopes are the pointwise maximum and minimum of the simulated values.

The significance level of the associated Monte Carlo test is alpha = 2 * nrank/(1 + nsim). The return value is an object of class "fv" containing the summary function for the data point pattern and the upper and lower simulation envelopes. It can be plotted using plot.fv.

Arguments can be passed to the function fun through .... This makes it possible to select the edge correction used to calculate the summary statistic. See the Examples.

If Y is a fitted point process model, and simulate=NULL, then the model is simulated by running the Metropolis-Hastings algorithm rmh. Complete control over this algorithm is provided by the arguments start and control which are passed to rmh.

Value

  • An object of class "fv", see fv.object, which can be plotted directly using plot.fv.

    Essentially a data frame containing columns

  • rthe vector of values of the argument $r$ at which the summary function fun has been estimated
  • obsvalues of the summary function for the data point pattern
  • lolower envelope of simulations
  • hiupper envelope of simulations

References

Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.

Diggle, P.J. Statistical analysis of spatial point patterns. Arnold, 2003.

Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.

Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.

See Also

fv.object, plot.fv, Kest, Gest, Fest, Jest, pcf, ppp, ppm

Aliases
  • envelope
Examples
X <- rpoispp(42)

 # Envelope of K function under CSR
 plot(envelope(X))
 <testonly>plot(envelope(X, nsim=5))</testonly>

 # Translation edge correction (this is also FASTER):
 plot(envelope(X, correction="translate"))
 <testonly>plot(envelope(X, nsim=5, correction="translate"))</testonly>

 # Envelope of K function for simulations from model 
 data(cells)
 fit <- ppm(cells, ~1, Strauss(0.05))
 plot(envelope(fit))
 <testonly>plot(envelope(fit, nsim=4))</testonly>

 # Envelope of G function under CSR
 plot(envelope(X, Gest))
 <testonly>plot(envelope(X, Gest, nsim=5))</testonly>

 # Use of `simulate'
 plot(envelope(X, Gest, simulate=expression(runifpoint(42))))
 plot(envelope(X, Gest, simulate=expression(rMaternI(100,0.02))))
 <testonly>plot(envelope(X, Gest, simulate=expression(runifpoint(42)), nsim=5))
  plot(envelope(X, Gest, simulate=expression(rMaternI(100, 0.02)), nsim=5))</testonly>
Documentation reproduced from package spatstat, version 1.7-11, License: GPL version 2 or newer

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